Rewrite the limand as
(1 - sin(x)) / cot²(x) = (1 - sin(x)) / (cos²(x) / sin²(x))
… = ((1 - sin(x)) sin²(x)) / cos²(x)
Recall the Pythagorean identity,
sin²(x) + cos²(x) = 1
Then
(1 - sin(x)) / cot²(x) = ((1 - sin(x)) sin²(x)) / (1 - sin²(x))
Factorize the denominator; it's a difference of squares, so
1 - sin²(x) = (1 - sin(x)) (1 + sin(x))
Cancel the common factor of 1 - sin(x) in the numerator and denominator:
(1 - sin(x)) / cot²(x) = sin²(x) / (1 + sin(x))
Now the limand is continuous at x = π/2, so
[tex]\displaystyle\lim_{x\to\frac\pi2}\frac{1-\sin(x)}{\cot^2(x)}=\lim_{x\to\frac\pi2}\frac{\sin^2(x)}{1+\sin(x)}=\frac{\sin^2\left(\frac\pi2\right)}{1+\sin\left(\frac\pi2\right)}=\boxed{\frac12}[/tex]
